c ESO 2010
Astronomy & Astrophysics manuscript no. 15580yu
September 14, 2010
Modelling the radio pulses of an ultracool dwarf
S. Yu1 , G. Hallinan2,3 , J.G. Doyle1 , A.L. MacKinnon4 , A. Antonova5 , A. Kuznetsov1,6 , A. Golden2 , Z.H. Zhang7
1
2
3
4
5
6
7
Armagh Observatory, College Hill, Armagh BT61 9DG, N. Ireland
e-mail: syu@arm.ac.uk
Centre for Astronomy, National University of Ireland, Galway, Ireland
Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA
Department of Adult and Continuing Education, University of Glasgow, Scotland
Department of Astronomy, Faculty of Physics, St Kliment Ohridski, University of Sofia, Bulgaria
Institute of Solar-Terrestrial Physics, Irkutsk 664033, Russia
Centre for Astrophysics Research, Science and Technology Research Institute, University of Hertfordshrie, Hatfield AL10 9AB
Received ; accepted
ABSTRACT
Context. Recently, unanticipated magnetic activity in ultracool dwarfs (UCDs, spectral classes later than M7) has emerged from a
number of radio observations. The highly (up to 100%) circularly polarized nature and high brightness temperature of the emission
have been interpreted as requiring an effective amplification mechanism of the high-frequency electromagnetic waves − the electron
cyclotron maser instability (ECMI).
Aims. We aim to understand the magnetic topology and the properties of the radio emitting region and associated plasmas in these
ultracool dwarfs, interpreting the origin of radio pulses and their radiation mechanism.
Methods. An active region model was built, based on the rotation of the UCD and the ECMI mechanism.
Results. The high degree of variability in the brightness and the diverse profile of pulses can be interpreted in terms of a large-scale
hot active region with extended magnetic structure existing in the magnetosphere of TVLM 513-46546. We suggest the time profile
of the radio light curve is in the form of power law in the model. Combining the analysis of the data and our simulation, we can
determine the loss-cone electrons have a density in the range of 1.25 × 105 − 5 × 105 cm−3 and temperature between 107 and 5 × 107
K. The active region has a size < 1 RJup , while the pulses produced by the ECMI mechanism are from a much more compact region
(e.g. ∼0.007 RJup ). A surface magnetic field strength of ≈7000 G is predicted.
Conclusions. The active region model is applied to the radio emission from TVLM 513-46546, in which the ECMI mechanism
is responsible for the radio bursts from the magnetic tubes and the rotation of the dwarf can modulate the integral of flux with
respect to time. The radio emitting region consists of complicated substructures. With this model, we can determine the nature (e.g.
size, temperature, density) of the radio emitting region and plasma. The magnetic topology can also be constrained. We compare
our predicted X-ray flux with Chandra X-ray observation of TVLM 513-46546. Although the X-ray detection is only marginally
significant, our predicted flux is significantly lower than the observed flux. Further multi-wavelength observations will help us better
understand the magnetic field structure and plasma behavior on the ultracool dwarf.
Key words. magnetic fields - radio continuum: stars - stars: low-mass, brown dwarfs - polarization - masers
1. Introduction
A large number of recent radio observations indicate that intense
magnetic activity exists in ultracool dwarfs (UCDs), i.e. objects
with spectral type later than M7. Berger et al. (2001) reported
the first detection of quiescent and flaring radio emission from
the M9 brown dwarf LP944-20, in which a bright X-ray flare
was also detected (Rutledge et al. 2000), with anomalous quiescent radio luminosity at least four orders of magnitude larger
than predicted from an empirical relation between the X-ray and
radio luminosities of active stars with spectral types from F to
M (Guedel & Benz 1993). The detection of electron cyclotron
maser (ECM) emission provided the first confirmation of kilogauss fields for a late M dwarf (Hallinan et al. 2006), and subsequently led to the discvery that even cooler L type dwarfs can
also possess magnetic fields in the kilogauss range (Hallinan
et al. 2008). Radio observations exclusively enable the measurement of magnetic fields on cool, brown dwarfs, and possibly also
for the very faint exoplanets (Zarka 2007).
Chromospheric Hα emission and coronal X-ray emission
show a sharp decline in LHα /Lbol and LX /Lbol beyond spectral
type M7 (Neuhäuser et al. 1999; Gizis et al. 2000; West et al.
2004; Stelzer et al. 2006a; Schmidt et al. 2007) which would
be consistent with lower fractional ionization in atmospheres of
later spectral type (Mohanty et al. 2002). Recently, however, a
lot of evidence, such as quiescent and flaring Hα emission from
some L and T dwarfs (Reid et al. 1999; Burgasser et al. 2000;
Liebert et al. 2003; Reiners & Basri 2007; Rockenfeller et al.
2006; Stelzer et al. 2006b; Schmidt et al. 2007), FeH lines from
cool M dwarfs (Afram et al. 2009), strong X-ray emission from
one such source (Audard et al. 2007), all suggest that perhaps an
efficient magnetic dynamo could be operational (Parker 1955) in
a fraction of the UCD population (Reiners & Basri 2010) and
that magnetic reconnection events could occur on these kinds
of cool objects. Such magnetic activity and thus the strong radio emission could be associated with the differential rotation
between the atmosphere and the core of the UCD.
Berger (2002) reported Very Large Array (VLA) observations of 12 late M and L dwarfs in the solar neighborhood.
Flare-like outbursts as well as persistent quiescent emission
were detected from three of the 12 sources, TVLM 513-46546
2
Yu et al.: Modelling the radio pulses of an ultracool dwarf
(TVLM 513 hereafter), 2MASS J00361617+1821104 (2MASS
J0036+18) and BRI 0021-0214. Among these three radio-active
sources, TVLM 513 and BRI 0021, plus the first detected
radio-active UCD, LP944-20, all have rapid rotational velocities,
v sin i >30 km s−1 . Hallinan et al. (2006) presented observations
of the rapidly rotating M9 dwarf TVLM 513 obtained simultaneously at 4.88 and 8.44 GHz using the VLA. The periodic radio
emission at both frequencies indicated a period of ∼2 hr in excellent agreement with the estimated period of rotation of the dwarf
based on its v sin i of ∼60 km s−1 . One more radio-active source
from Hallinan et al. (2008), LSR J1835+3259, also has a high
rotational velocity, v sin i ≈50 km s−1 .
Up to now, about 10 radio active ultracool dwarfs, with spectral type from M8 to L3.5, including a binary system (2MASS
J07464256+2000321, hereafter 2MASS 0746+20), have been
found from various surveys (Berger 2002; Burgasser & Putman
2005; Berger 2006; Hallinan et al. 2006, 2007, 2008; Antonova
et al. 2008). Three of these have been shown to have periodic radio emission, with periods of 1.96 hr (TVLM 513, Hallinan et al.
(2007)), 3.07 hr (2MASS J0036+18, Hallinan et al. (2008)), and
2.83 hr (LSR 1835+32, Hallinan et al. (2008)).
The L dwarf binary, 2MASS 0746+20, reported by
Antonova et al. (2008) from a mini-survey of UCDs at 4.9 GHz
has a high mean flux level of 286±24 µJy. Berger et al. (2009)
presented an 8.5 hr simultaneous radio, X-ray, UV, and optical
observation of this binary. The strong radio emission consists
mainly of short-duration periodic pulses at 4.86 GHz with P =
124.32 ± 0.11 min. The radio pulses are 1/4 phase different from
the Hα emission.
The narrow bunching of multiple pulses of both left- and
right- 100% polarized radio emission detected from TVLM 513,
which originate in regions of opposite magnetic polarity, reveal
the likely presence of a dipolar component to the large-scale
magnetic field (Hallinan et al. 2008). Zeeman Doppler Imaging
(ZDI) observations have also shown that such a large-scale dipolar magnetic structure could exist on an M4 dwarf star, V374
Peg, which is also a fully convective rapid rotator similar to
TVLM 513 (Donati et al. 2006; Morin et al. 2008). The topology of magnetic fields on UCDs needs to be constrained by more
observations.
The radio emission composed of quiescent and pulsing components from UCDs can be associated with not only the geometry of the emitting region and rotation of UCDs, but also the
behavior of plasma in the magnetic field, i.e. the radiation mechanism. The very high brightness temperature and high (up to
100%) circular polarization of the pulses (Hallinan et al. 2007)
point towards an efficient, coherent radiation mechanism, the
electron cyclotron maser instability (ECMI, Melrose & Dulk
(1982)). This wave magnification process of the free-space radiation modes could be induced by some kind of anisotropic velocity distributions of electrons, such as a loss-cone distribution
(Lau & Chu 1983), ring shell distribution, or horseshoe distribution (Pritchett 1984).
ECMI was successfully applied to the auroral kilometric radiation (AKR) on Earth (Wu & Lee 1979; Ergun et al. 2000), decametric radiation (DAM) on Jupiter, Saturnian kilometric radiation (SKR) (Zarka 1998, 2004) and solar millisecond microwave
spikes (Aschwanden 1990b). Various authors have suggested its
presence in exoplanets (Zarka 2007; Grießmeier et al. 2007;
Jardine & Cameron 2008). The magnitude of any contribution
from incoherent gyrosynchrotron or synchrotron radiation to the
quiescent components in UCDs is uncertain.
The generation of ECMI emission is dependent on the environment of the emitting region, such as the magnetic field
(strength, structure), the electron density distribution (number,
energy), the line-of-sight source scale, and the angle between
the line of sight and the magnetic field (Melrose & Dulk 1982),
and also the details (velocity space gradients) of the loss cone
distribution (Aschwanden 1990a). This results in the possibility of transient radio emission. The findings of Antonova et al.
(2007) indicate that UCDs may also have sporadic long-term
variability in their levels of quiescent radio activity. This phenomenon could be related to the change of the environment
of the radio-emitting region, leading to self-quenching of the
electron-cyclotron maser. Radio observations can help us to determine the magnetic configuration and the nature of the plasma
in the emitting region.
For the purpose of understanding the magnetic topology, we
built an active region model based on the rotation of TVLM 513
and the ECMI mechanism to simulate the observed light curve.
We summarize the previous radio observation on TVLM 513 in
§2. In §3, we present the model, and results are given in §4. We
discuss these results in §5 and draw conclusions in §6.
2. Previous observations on TVLM 513
TVLM 513 is a young radio active M8.5V dwarf with a bolometric magnitude of log(Lbol /L ) ≈ −3.65, effective temperature
T eff ≈ 2200K (Tinney et al. 1993, 1995; Leggett et al. 2001), and
situated at a distance of d = 10.6 pc (Dahn et al. 2002). From the
theory of the formation and evolution of UCDs and the absence
of lithium on TVLM 513, it is reasonable to infer values for the
mass and radius of this star of ∼0.07 M and ∼0.1R respectively
(Reid et al. 2002; Chabrier & Baraffe 2000).
With the VLA a highly right-circularly polarized (∼65%) radio event from TVLM 513 was detected with a flux density of
∼1100 µJy, as well as persistent variable emission at 8.46 GHz
(Berger (2002)). Osten et al. (2006) conducted a multifrequency
VLA observation of TVLM 513 at 8.4, 4.8 and 1.4 GHz, using
a strategy that involved time-sharing a single 10 hr observation
between the various frequency bands. TVLM 513 was detected
at each frequency band with only marginal confirmation of variability and no detection of flares or strong circular polarization.
Again, using the VLA, Hallinan et al. (2006) found persistent
and periodic radio emission from TVLM 513 at 8.44 GHz and
4.88 GHz simultaneously, with a period of ∼2 hr. Subsequently,
extremely regular periodic bursts (p = 1.96 hr, up to ∼4 mJy) of
high brightness and highly circularly polarized radio emission
were reported by Hallinan et al. (2007). Multiple bursts of both
left and right 100% circularly polarized emission were detected.
Interestingly, the radio emission can switch states from nearly
100% left polarization to 100% right polarization in each phase.
Another radio burst with a flux density up to ∼4 mJy was
presented by Berger et al. (2008) from a period of simultaneous radio, X-ray, ultraviolet, and optical spectroscopic observations. Steady quiescent radio emission superposed with multiple,
short-duration, highly polarized bursts was observed, but these
authors reported a non-periodicity in the pulses/flaring activity.
In a re-analysis of this data, plus data taken ≈40 days later (June
2007), Doyle et al. (2010) reported the 1.96 hr periodicity in both
datasets, deriving a more accurate period.
3. Model
In this section, we present a model (Figure 1) to simulate the
observational time profile of the radio flux density from TVLM
513 by assuming ECM emission is the dominant radiation mechanism.
Yu et al.: Modelling the radio pulses of an ultracool dwarf
3
beam could result from the interaction of a close-in companion
of the UCD, i.e. magnetized or non-magnetized satellites resulting in auroral emission, similar to Io-Jupiter system (Queinnec
& Zarka 1998; Saur et al. 2004; Zarka et al. 2005), is possible. ECM emission has been detected in compact objects, such
as white dwarfs (Willes & Wu 2004, 2005) or neutron stars
(Wolszczan & Frail 1992). We are not able to rule out this model
and the competition between the two models should be investigated in further work.
3.1. Flux calculation
The fine structure and time interval of the observations described
in §2 indicate that the radio-emitting region on TVLM 513
would consist of complex substructures. The detected flux density S̃ may be the sum of several small sources
Fig. 1. A sketch of the active region model on ultracool dwarfs.
S̃ =
ñ(t)
X
S i.
(1)
i=0
When plasma electrons are energized in magnetic flux tubes
with converging legs and foot-points in a high density atmosphere (perhaps as a consequence of magnetic reconnection),
some of these fast electrons collide with the high density atmosphere and thermalise at the foot-points. The remaining fast
electrons are reflected in the converging field by a magnetic mirror effect. This process results in the formation of the anisotropic
distribution of the plasma in velocity space, i.e. a loss-cone distribution.
The plasma including a loss-cone distribution is unstable; instability arises very quickly from such a distribution. A large
amount of free energy can be released via the instability and
converted to electromagnetic waves - see Dulk (1985) for a review of the process. An external magnetic-field-aligned electric
field (Cattell et al. 1998; Zarka 1998; Ergun et al. 2000) induced
by a time-varying external current source (Omura et al. 2003)
would further modify the plasma velocity distribution to shell or
horseshoe form, leading to an enhanced ECMI emission. In this
paper, we assume for simplicity that the ECMI emission from
UCDs is driven by the loss-cone distribution. The existence of
the electric field and its effect will be addressed in future work.
In a loss-cone region where the ECMI operates, the maser
radiation is concentrated on the surface of a hollow cone as
discussed by Melrose & Dulk (1982) (see Fig. 1 top panel).
The half-angle α0 of the hollow cone depends on the ratio of
the velocity of the plasma electrons to the speed of light, i.e.
cosα0 = v p /c. For example, if v p /c = 0.5, we have α0 = 60◦ .
The surface of the cone should be very thin with ∆α ≈ v p /c.
The maser emission in the loss-cone region can be observed if
the line of sight is located within a thin conical sheet with thickness ∆α. For a low magnetic loop with a small angle between the
magnetic field and the surface of the UCD, the maser emission
can be seen when the emission is near the top of the loop. The
maser emission from near the foot-point (where the magnetic
field is almost perpendicular to the surface of UCD) can be seen
when the loop is near the limb. However, for a large scale, the
maser emission could have an angular distribution. In our calculation, we assume we can observe the maser emission in each
flux tube in the active region.
By analogy to the solar coronal radio emission powered by
two populations of plasma from the Sun with different velocity,
we propose there be similar active regions on UCDs. We note
that, however, an alternative mechanism, where the hot plasma
where S i is the flux density for a small source which can be
determined by the relation (Dulk 1985)
!2
Z
f
T b dΩ
(2)
Si =
kB
c
where T b is the brightness temperature of a source, kB is the
Boltzmann constant, f is the observed frequency, c is the speed
of light, dΩ is the differential solid angle. If we assume that the
radiation is isotropic, the differential solid angle should depend
on the radius of the flux tube rtube of the small source and its
distance from the observer. The flux density S i can be expressed
as
!2
πr2
f
S i = kB
(3)
T b tube2 .
c
4πd
For simplicity, we assume that the active region has a symmetric shape, and the small sources are randomly distributed
within the region. The number of small sources ñ(t) may be determined by rotation of the UCD. We set the time t = 0 to be
the moment when the first active region emerges in the field of
view of the observer; this is also the onset time of a radio pulse.
With the rotation of the UCD, the area of the active region seen
by the observer increases until it reaches a maximum, and then it
decreases and disappears from the field of view of the observer.
The maximum number of small sources is
!γ
∆t · 2πR0 cos θ/T UCD
ñmax ≈
.
(4)
rtube
where T UCD is the rotation period, R0 is the height of the active
region, θ is the latitude of the active region and ∆t is the time
interval from the beginning of a pulse to the maximum flux. The
time duration of a pulse would be 2∆t for a symmetric shape. We
assume that the radio emission is from a thin shell of the active
region near the surface of the UCD, so that we can take γ ≈ 2
and R0 ≈ RUCD where RUCD is the radius of the UCD.
3.2. Brightness temperature and relative parameters
The brightness temperature of small sources depends strongly
on the growth rate of the ECM emission and incoherent radiation of the background plasma. If we assume the maser emission is operated by the loss-cone distribution of a population of
4
Yu et al.: Modelling the radio pulses of an ultracool dwarf
Fig. 2. The change of growth rate and brightness temperature with different plasma parameters and loss cone parameters. fp / fc = ratio of plasma
frequency and gyrocyclotron frequency, T w = background wave energy, nh = density of hot plasma, T h = temperature of hot plasma, αc = loss cone
angle, N = loss cone pitch angle distribution slope, T b = brightness temperature. The range of each parameter is in the definition of Aschwanden
(1990a). We plot each panel assuming ’standard’ values of the other parameters (see Table 1).
hot plasma expressed by a Maxwellian multiplied by the function sinN (α/αc · π/2) and an isotropic Maxwellian distribution
of a cold background plasma, the brightness temperature can be
determined by the parameters of these two types of plasma. We
adopt the quasi-linear theory developed by Aschwanden (1990a)
to determine the growth rate and the efficiency of energy conversion. Here, we summarize the basic assumptions and the effects
of several free parameters.
The quasi-linear code of Aschwanden (1990b) describes the
evolution of the ECM instability and the wave-particle inter-
actions by solving the kinetic wave-particle equations in a locally homogeneous plasma. The wave equation includes induced gyroresonance emission/absorption (for the X-, O-, Zmagneto-ionic modes and whistlers), but neglects spontaneous
emission, free-free absorption, collisional deflection, and spatial
wave propagation. The coupled diffusion equation contains the
quasi-linear diffusion coefficients due to maser growth/damping,
but neglects slower processes like particle loss and source terms.
This assumption corresponds to the strong diffusion case. The
quasi-linear diffusion process of the ECM instability in the
Yu et al.: Modelling the radio pulses of an ultracool dwarf
Table 1. Parameters to determine the flux density. Plasma parameters:
nh = density of hot plasma, T h = temperature of hot plasma, u = fp / fc
= ratio of plasma frequency and gyrocyclotron frequency, nc = density
of cold (background) plasma, T c = temperature of cold (background)
plasma, T w = background wave energy; Loss cone parameters: αc = loss
cone angle, N = loss cone pitch angle distribution slope, B = magnetic
field strength; UCD parameters: RUCD = radius of UCD, T UCD = rotation
period of UCD, rtube = radius of flux tube, θ = latitude of radio-emitting
region, A = size of emitting region. Note that the number density of cold
plasma is associated with fp / fc .
Plasma
Loss-cone
UCD
nh
(cm−3 )
1.25×105
αc
(degree)
30
RUCD
(km)
7.1×104
Th
(K)
107
N
6
T UCD
(hr)
1.96
u ( fp / fc )
0.1
B
(G)
1750
rtube
(km)
55
Tc
(K)
106
θ
(degree)
30
Tw
(K)
1014
A
(km2 )
8202
solar corona successfully accounted for the time profile and
observational characteristics of decimetric millisecond spikes
(Aschwanden 1990b).
In order to constrain the growth rate Γ, energy conversion
factor εc and saturation of the maser, we need to know the initial
nature of the two populations of plasma and the shape of the
loss-cone, i.e. for the hot plasma: particle density nh , particle
temperature T h ; for the cold plasma: particle density nc , particle
temperature T c ; for the loss-cone: loss-cone angle αc , pitch angle
distribution slope N. Rather than choowing values of the cold
plasma density nc we fix its value via adoption of a value for
u = fp / fc , the ratio of plasma frequency ( fp ' 9 × 10−3 (nc )1/2
MHz) and gyrofrequency ( fc ' 2.86 × B MHz) where B is the
magnetic field strength in G.
In the quasi-linear diffusion process, the existence of background electromagnetic wave energy T w is taken into account.
As discussed by Aschwanden (1990a) in the case of solar millisecond spikes, the initial brightness temperature could be as
low as the level of thermal bremsstrahlung in the range of 106 ∼
108 K. However, because of gyroresonance or gyrosynchrotron
radiation, an enhanced photon level could exist in a flaring loop
which would affect the ECM process significantly, yielding a
wave turbulence at the level of ∼ 1015 K. We apply T w in the
range of 106 − 1016 K in our model.
The free energy of the plasma with a loss-cone distribution
can be converted into an equivalent electromagnetic energy. The
energy conversion factor εc can be defined as the ratio of the
change of kinetic energy between the initial state and the final
state of the plasma and the initial kinetic energy. In the present
paper, the amount of converted energy is about 0.5%, i.e. the
same as in Aschwanden (1990a).
Aschwanden (1990a) investigated how the parameters influence the growth rate and brightness temperature generated by
the ECM instability and gives the general formula for the key
parameters as follows:
5
α α −1 !−1
nh
c
c
−1.1 + 1.1
+
Γ = 6.9 × 10
30◦
30◦
1.25 × 106 cm−3
T T −1.5 !−1 N
0.45
h
h
× 0.65 + 0.05
+
0.30
1.15
−
0.15
6
108 K
108 K
2 
u
u


+ 0.2 0.1
,
0.1 < u 6 0.24
2.0 − 1.2 0.1




2

u
×
0.24 < u 6 1.0
0.19(1.0092 − 0.0092 0.1 ),



2


0.021(−1.416 + 0.440 u − 0.024 u ), 1.0 < u < 1.4
0.1
0.1
(5)
5
T b = 9.0 × 1017
1.10
α −1 !
nh
c
1.4
−
0.4
30◦
1.25 × 106 cm−3
!
!−0.05
N 0.3
log T w
2.0
− 1.0
6
14
(6)
0.1 < u 6 0.24
0.24 < u 6 1.0
T 1.2
h
108 K



1,



9,
×

−3



1400 u
,
0.1
×
1.0 < u < 1.4
We plot in Figure 2 the effect of various parameters on the
growth rate and brightness temperature.
4. Results and comparison with observations
Many parameters, not only those describing the nature of the
plasma but also those associated with the properties of the UCD,
can affect the observable flux density significantly. In this section, we will discuss these parameters and compare our simulations with observations. In order to study the influence of the
various physical quantities we adopt ‘standard’ values for each
parameter, listed in Table 1. Unless otherwise noted, the simulations are always for magneto-ionic X-mode σ = −1, harmonic
number s =1.
4.1. The effect of rotation
In order to see the effect of rotation of the UCD, we first start
a set of simulations by fixing the initial plasma and loss-cone
parameters as listed in Table 1. In addition, as suggested by
Chabrier & Baraffe (2000), we adopt the radius (RUCD =0.1 R )
as a constant in our simulations. We initially assume that the
radio-emitting region is close to the equator (latitude θ = 30◦ ).
Given the above parameters, the size (A) of the emitting region and the tube size of a small radio area, we show in Fig. 3 the
influence of rotation. From the figure, we can see that the radio
light curve is broadened with increasing rotation period (T UCD )
while the intensity does not change at all. This is because the
intensity of the radio emission is only related to the behavior of
plasma and the total size of the emitting region.
Figure 4 shows the effect of the total size (A) of the radioemitting region (left panel) and the radius (rtube ) of the flux tube
(right panel). It is easy to understand that rtube cannot change the
radio flux as much as T UCD . It could, however, be a controlling
factor for the observed oscillation of radio flux because each flux
tube could have a different environment so that we may have to
give a distribution for the initial plasma parameters (see §4.5).
A is an important parameter for the nature of the radioemitting region on the UCD, since we can easily see from Fig.
6
Yu et al.: Modelling the radio pulses of an ultracool dwarf
4.3. The effect of cold plasma and the initial wave energy
The number density nc of the cold plasma can be obtained from
the ratio of the plasma frequency to the cyclotron frequency
( fp / fc ). Given a magnetic field strength (B=1750 G), fc =4.9
GHz. Since the standard value of fp / fc =0.1, fp =490 MHz. This
implies nc = 3×109 cm−3 , the standard value adopted in our simulations. Note from Fig. 2, that changing fp / fc from say 0.1 to
0.2 makes no difference to the brightness temperature.
0.003
T
T
T
=0.96 hr
UCD
=1.96 hr
UCD
=2.96 hr
UCD
Flux (Jy)
0.002
In panel (a) of Fig. 5 we show the effect of changing fp / fc
for a constant B. We see that the flux density goes up a little
from fp / fc =0.23 to 0.5 (nc =1.6×1010 to 7.4×1010 cm−3 ), and
then drops when fp / fc is increased further to 1.2 (nc =4.3×1011
cm−3 ). This result is consistent with the variation of the brightness temperature in Fig. 2 because the flux density in our simulation depends strongly on the brightness temperature.
0.001
0.000
0
10
20
30
40
50
t (s)
Fig. 3. The influence of rotation period (T UCD ) on the radio light curve.
Vertical scale denotes the flux density while the horizontal scale indicates the time. Dashed line, solid line and dotted line are for T UCD 2.96,
1.96, 0.96 hr, respectively.
4 that increasing its value can lead to a rise in both intensity and
pulse duration. A can be constrained by the observations once
we know the rotation period of the UCD. We are able to evaluate
A from
A ∼ (∆t · 2πRUCD cos θ/T UCD )γ
(7)
where the parameters are the same as in Eq. 4. For the standard
parameters and γ = 2, we find ∆t ≈ 15 s. This implies that the
observed ∆t can be used to estimate A.
4.2. The effect of hot plasma in the loss-cone
Because of the conservation of energy, the electromagnetic wave
energy escaping from the radio- emitting region has come from
the kinetic energy of the hot plasma. So the brightness temperature of one flux tube should be proportional to the number density and temperature of hot particles.
We can see in panels (c) and (d) in Fig. 5 the variation of the
flux density (S ) with the number density (nh ) and temperature
(T h ) of the hot plasma. Simulations in this section assume standard values for all parameters except nh and T h . Increasing T h
from 107 to 108 K, S will increase by ∼16 times, while S rises
by a factor of 12 when nh is one order of magnitude higher. This
is consistent with the change in the brightness temperature (see
Fig. 2).
It is easy to understand that the population of hot electrons
is shifted to higher velocities where the number of undamped
resonance ellipses increases and thus the growth rate increases
with increase in temperature. On the other hand the cold background plasma will negate the loss cone and the growth rate
will decrease sharply due to the lack of undamped resonance
ellipses, if the hot electron temperature approaches that of the
cold plasma. Melrose et al. (1984) gives a criterion for effective cyclotron damping by background cold electrons, which is
T h /T c < 10 ∼ 20. This value was confirmed in the work of
Aschwanden (1990a).
We note, however, that, although the growth rate declines
persistently in Fig. 2, the brightness temperature increases
steeply by almost one order of magnitude when the magnetoionic mode changes from X-mode to O-mode at harmonic number s=1. This is because of the Doppler resonance condition as
discussed by Aschwanden (1990a). The resonance ellipses covering the unstable portion of the loss-cone distribution form a
smaller region of positive growth for the O-mode, and a higher
wave level results from the same amount of energy conversion.
We need to mention that only the dominant magneto-ionic
modes, i.e. fundamental (s=1) X-mode when fp / fc < 0.24, fundamental (s=1) O-mode when 0.24 < fp / fc < 1.0, and second
harmonic (s=2) X-mode when 1.0 < fp / fc < 1.4, have been
taken into account in our simulations. When fp / fc > 1.4, some
electrostatic instabilities become important rather than the ECM
instability, and a condition for ECM emission escaping from
plasma is fp / fc 1. The fundamental (s=1) Z-mode may be
dominant when fp / fc ≈ 0.3 and fp / fc ≈ 1.1 (Melrose et al.
1984; Aschwanden 1990a). However, we omit this mode because its appearance depends strongly on the ratio fp / fc and another mechanism would be needed to convert it to electromagnetic radiation.
When T w (defined as the initial wave energy) results mainly
from thermal bremsstrahlung it takes a low value of 106 ∼ 108
K. T w may become larger, in the range of 1010 ∼ 1016 K, due to
incoherent gyroresonance or gyrosynchrotron radiation. Lower
T w can lead to a higher energy conversion efficiency because
of the effect on the resonance ellipses. The flux density drops
slightly with increasing T w as seen in panel (b) of Fig. 5.
4.4. The effect of loss-cone parameters
Panel (e) in Fig. 5 shows the effect of the loss-cone angle on
flux density. The rapid increase of flux density when the losscone angle (αc ) changes from 10◦ to 20◦ is the consequence
of the significant increase in the size of the region of velocity
space involved. Quasi-linear diffusion influences a large number of particles and results in growth and amplification of some
wave modes growing. On the other hand, when αc > 20◦ , the
positive gradient in the perpendicular direction decreases so that
the electromagnetic wave energy density can not be amplified
effectively.
The effect of pitch angle distribution slope is similar to that
of the loss-cone angle, see panel (f) in Fig. 5.
Yu et al.: Modelling the radio pulses of an ultracool dwarf
A=4402 km2
A=8202 km2
A=12002 km2
0.006
0.005
7
rtube=15 km
rtube=55 km
rtube=95 km
0.003
0.004
Flux (Jy)
Flux (Jy)
0.002
0.003
0.002
0.001
0.001
0.000
0.000
0
10
20
30
40
50
60
70
0
5
10
15
t (s)
20
25
30
35
40
t (s)
Fig. 4. Flux density against time with different size of the emission region (A, left panel) and radius (rtube , right panel) of one flux tube.
0.00272
(b)
(a)
0.025
O
0.00268
s=1
0.015
flux (Jy)
0.3
flux (Jy)
flux (Jy)
0.020
(c)
0.4
0.00264
0.2
0.010
X
X
s=1
0.005
0.00260
0.1
s=2
0.000
0.00256
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.0
6
8
10
fp / fc
0.30
T
log
0.003
(d)
12
14
16
0
2
4
6
8
10
12
nh (106cm-3)
w (K)
(e)
(f)
0.003
0.25
0.15
0.10
flux (Jy)
0.002
flux (Jy)
flux (Jy)
0.20
0.002
0.001
0.001
0.05
0.00
0.000
0
1
2
3
Th (108 K)
4
5
0.000
10
20
30
c (degree)
40
50
0
2
4
N
6
8
10
Fig. 5. Simulated maximum flux density of a radio pulse as a function of the ratio of cyclotron frequency and plasma frequency ( fp / fc ), background
wave energy (T w ), density of hot plasma (nh ), temperature of hot plasma (T h ), loss cone angle (αc ), loss cone pitch angle distribution slope (N) are
shown in panel (a) to (f) respectively. The parameters for the calculation in this plot are the same as in Fig. 2.
8
Yu et al.: Modelling the radio pulses of an ultracool dwarf
4.5. Determining the environment of the radio-emitting region
In this section, we determine the possible range of the different
parameters. From Table 1, there are at least 13 parameters that
can influence the radio light curve of TVLM 513; we are able to
reduce this number, however.
Considering the contribution of the degenerate electron gas
and the ionic Coulomb pressure, the radius is almost a constant
around 0.1R , in the range of 0.08R to 0.11R (Chabrier &
Baraffe 2000).
The temperature of cold plasma is more difficult to determine. In this work we assume it is around 106 K, similar to the
typical temperature of the solar corona. On the other hand, if
the coronal temperature was lower, this would favor wave propagation rather than damping. We set the initial wave energy T w
at a high level of 1014 K. On the other hand, from Fig. 5, the
flux density varies only slightly (a factor of 2) when T w changes
from 1010 to 1015 K. We should however note that Hallinan
et al. (2008) suggested that the ECM instability may be a viable
source of quiescent unpolarized radio emission, indistinguishable in temporal and polarization characteristics from gyrosynchrotron radiation.
For convenience we adopt a plausible value of 0.1 for the
ratio u of plasma frequency to gyrofrequency ( fp / fc ), a value
that permits unstable growth of the ECMI. If we focus on the
observation frequency of 4.9 GHz, the magnetic field strength
(B) has to be around 1750 G and the density of the cold background plasma is ∼3×109 cm−3 . Any change of u alone will not
affect the maximum flux density for the fundamental X-mode
and fundamental O-mode respectively. However, we stress that
if we change the magnetic field strength (B), even keeping u at
the same value, the maximum flux density will change because
it depends on the cyclotron frequency.
The size of the radio-emitting region A is associated with the
time duration of the radio pulses. Thus in the case of TVLM 513,
we have A ≈ (54.75∆t)γ when the latitude of the active region
is 30◦ . Changing the latitude to 70◦ gives A ≈ (21.62∆t)γ . ∆t is
easily obtained from observations and is a more convenient parameter than A. rtube is a variable to describe the radius of a flux
tube. We expect there are distributions for plasma parameters
(T h , nh ) and loss-cone parameters (αc , N) in different flux tubes.
The variation of different parameters can result in the oscillation of the observed flux density. Smaller values for rtube lead to
smoother simulated radio light curves (see Fig. 4).
So now the free parameters are reduced to 6, i.e. the density
nh and temperature T h of hot plasma, the angle αc and pitch angle
distribution slope N of the loss-cone, the radius rtube of one flux
tube and the size of the emitting region A. The functions of the
parameters are: nh and T h can control the change of flux density
dramatically, αc and N can control the change of flux density
gently, A can control both flux density and time duration of the
radio pulse which can be constrained by observations, and rtube
describes the radius of the flux tube in the active region.
4.6. Comparison with observations
The free parameters can be constrained by making our simulations match up to observations. Fig. 6 shows the comparison
of our simulation with observations (see Hallinan et al. 2010
for further details). The observation is taken from 2008 May
19 (UT) at 4725 MHz for the M8.5V dwarf TVLM 513. This
means that the magnetic field strength is 1652 G because of the
cyclotron frequency. We adopt fp / fc = 0.1, implying the density
of cold plasma in the emitting region is ∼2.64×109 cm−3 . The
size of the radio-emitting region is constrained by the time duration of the two pulses (A ≈ (54.75∆t)2 ), implying 1620×1620
km2 for the first pulse and 2060×2060 km2 for the second pulse.
The radius of each flux tube is taken as 180 km arbitrarily because we are not able to get any information on its value from
the observed radio light curve. We take αc = 30◦ and N = 5 for
the first pulse, αc = 35◦ and N = 2 for the second pulse. T h and
nh are same for the both pulses, equal to 107 K and 1.25 × 105
cm−3 respectively.
The simulated light curve reproduces the observation.
However, this does not mean that the above values for the different parameters are unique. In order to compare our simulation
with the observed two pulses, we take a flat distribution for αc
and N in the range of 10◦ − 50◦ and 1 − 6 for the both pulses,
nh : (1.25 − 5) × 105 cm−3 , T h : (1 − 4.5) × 107 K for the first
pulse and nh : (1.25 − 4.5) × 105 cm−3 , T h : (1 − 3) × 107 K
for the second pulse (see inset panel in Fig. 6). We note that the
size of the emitting region depends on the latitude θ in our simulation (see §4.5). Changing the latitude from 30◦ (the value for
the simulation) to 70◦ gives a value for the size of 640×640 km2
for the first pulse and 814×814 km2 for the second pulse. Other
parameters also need to change in order to fit the observed light
curve.
The simulation in Fig. 6 is remarkably similar to the observations. Our results indicate that the two radio emission regions
producing the two pulses are very close, which would be consistent with the nature of the cone radiation of ECMI. An important
feature is that the radio pulses would repeat with the rotation period of the UCD. In the case of TVLM 513, the period is ∼1.96
hrs. Moreover, we note that the decay time of the pulses in some
observations is longer than our simulation. This is possibly due
to the deformation of the radio-emitting region as fast rotation
of the dwarf can cause the shape of the emitting region to vary
from an almost symmetric circle to an asymmetric ellipse with a
tail.
5. Discussion
Our results show that rotation coupled with the ECMI mechanism can account for the flux density and polarization of the
radio pulses from TVLM 513 successfully. We can not exclude
the possibility that the depolarization of ECMI could be due to
radiation transfer of the emission in a neutral atmosphere with
lower fractional ionization or that inhomogeneous dust clouds
(Littlefair et al. 2008) could have an effect on the quiescent emission and the unpolarized components. Hallinan et al. (2006) suggested that the depolarization or mode conversion of the X-mode
emission occurs in a density cavity, as mode conversion of terrestrial kilometric radiation (TKR) from X-mode to R-mode in
the emitting density cavity (Ergun et al. 2000) may account for
escape of maser emission without re-absorption at higher harmonics of the emission frequency.
For the optically thick source, i.e. optical depths τ 1, the
brightness temperature T b can be constrained by Eq. 3 by using
the observed flux density. In order to see the relation between T b
and flux density, we rewrite the equation in the form of
!
!−2
!2
!−2
Sf
f
d
Rs
10
T b = 5.32 × 10 K
4 mJy 4.9 GHz
10 pc
RJup
(8)
where S f is the flux density in mJy at the frequency f (GHz), d
the distance of the radio source from us in pc, Rs the size of the
emission region in Jupiter radii (1 RJup ∼ 0.1 R ≈ 7 × 109 cm)
(Linsky & Gary 1983; Doyle et al. 1988; Dorman et al. 1989;
Yu et al.: Modelling the radio pulses of an ultracool dwarf
9
10
Relative flux (mJy)
12
Relative flux (mJy)
10
8
8
6
4
2
0
-2
6
11430
11460
11490
11520
11550
Time (s)
4
2
0
-2
11440
11460
11480
11500
11520
11540
11560
Time (s)
Fig. 6. Comparison of our simulation with observation taken with the Arecibo telescope by Hallinan et al. (2010). Black solid line shows the
observation of 2008 May 19 (UT) at 4775 MHz. Red line shows the simulated curve. The inset panel shows the simulated light curve using a
uniform distribution of the free parameters.
Burrows et al. 1989; Leto et al. 2000). Unfortunately, when we
calculate T b , we have to assume the size of the radio-emitting region. For example, observations show that the flux density of the
pulses of TVLM 513 is about 4 mJy at ∼4.9 GHz. Berger (2002)
obtained the brightness temperature in the range of 108 − 109 K
by assuming the size of a corona to be 2 − 4 RJup , while Hallinan
et al. (2006) deduced a value of 2.9×1010 K if the size of the
region is 1 RJup . In our model, T b & 5 × 1010 K is about the
temperature of the quiescent radio emission, assumed to come
from a large emission region (. 1 RJup ). For the radio pulses
from magnetic loops, the theoretical temperature of the coherent ECMI emission can be up to 1015 K, implying the emission
region for the pulses is much more compact, e.g. 0.007 RJup .
existence of a dipolar large-scale magnetic field (Hallinan et al.
2007) or a few small active regions with scale << 1 RJup .
Another important parameter, when combined with the magnetic field strength, is the pitch angle β. Electrons with small β
less than a critical value βc , i.e. β < βc , precipitating into the
dense atmosphere are lost, while the electrons with β > βc will
be reflected back to the flux tube to form an anisotropic velocity
distribution. The value of βc depends on the convergence factor
which is determined by the ratio of magnetic field strengths at
the top and the foot-points of the flux tube (also called magnetic
mirror ratio). For a symmetric flux tube, we have (Dulk 1985)
The configuration and topology of magnetic field on UCDs
remains unclear. In our model, we only need a simple dipole
poloidal-like field to calculate the flux density and explain the
high polarisation of the radio pulses. In the case of TVLM 513,
multiple bursts of both left and right 100% circularly polarized
emission in regions of opposite magnetic polarity indicate the
where Btop and Bfoot are the magnetic field strengths at the top
and in the foot-point of the magnetic flux tube, respectively.
Typical values of Btop /Bfoot in the Sun are in the range of 0.1 to
0.5. In the simulation, the pitch angle is approximately equal to
30◦ , which means we should have 0.5 > sin βc = (Btop /Bfoot )1/2 .
This could give a lower limit on Bfoot .
βc = arcsin(Btop /Bfoot )1/2
(9)
10
Yu et al.: Modelling the radio pulses of an ultracool dwarf
1.0
Dulk & McLean (1978)
for solar coronal magnetic field
0.8
B / B
0
0.6
Unipolar-like field
0.4
0.2
Eq. 10
0.0
0
1
2
3
4
5
6
7
8
9
R/ R
Fig. 7. Possible magnetic field structures for UCDs. The solid line is for
Eq. 10 which was suggested for a dM star, the dashed line for a unipolarlike field, the dotted line for the possible field in the solar coronal region.
Assuming the magnetic field is radial, the field strength of a
region can be described by
B0 R
1
B=
−
4 R∗ 2
!−2
, R > R∗
For the thermal bremsstrahlung emission of ionized hydrogen and helium dominated source, the detected flux
density integrated over frequency is S X−B = 4.67 ×
10−28 T h1/2 nh nc Z 2 gB R3s d−2 , where Z is an ion of charge in units
e (here we take Z = 1), gB the Gaunt factor (we take gB = 1.2,
which gives an accuracy of .20% since 1.1 < gB < 1.5), Rs
the radius of the source, d distance of the source from us and
all quantities are in cgs units (Rybicki & Lightman 1979). In the
case of TVLM 513, assuming that the X-ray emission comes
from the same region as the radio emission, we can take the
parameters as T h = 107 K, nh = 108 cm−3 , nc = 3 × 109
cm−3 , Rs = 2 × 108 cm, d = 10 pc≈ 3.1 × 1019 cm, hence
we get the S X−B = 4.42 × 10−21 erg·cm−2 ·s−1 , giving an X-ray
luminosity of LX−B ≈ 5.34 × 1019 erg·s−1 . The X-ray flux density/luminosity could be underestimated significantly as the Xray emission might be diffuse and from a larger region (perhaps
∼10 times) than that of the radio emission, as is the case for the
X-ray emission observed from Jupiter by S uzaku (Ezoe et al.
2010).
Interestingly, Berger et al. (2008) obtained a marginal detection in X-rays suggesting a flux density of 6.3 × 10−16
erg·cm−2 ·s−1 (luminosity LX = 8.5 × 1024 erg·s−1 ) with mean energy at 0.9 keV. This means the temperature of the hot plasma
could be slightly lower than 107 K. Further multi-wavelength
observations will help to refine our model and its parameters to
understand the radio and X-ray emission from these kinds of
cool objects.
(10)
where B0 is the magnetic field at the photosphere, R the height
of the radio emitting region measured from the centre of the star
and R∗ the photospheric radius. This was chosen as a compromise between a unipolar-like field B = B0 (R/R∗ )−2 and the bestfit form for solar coronal magnetic fields above active regions
B = 0.5[(R/R∗ ) − 1]−1.5 given by Dulk & McLean (1978) for the
range 1.01 < R/R∗ < 10. This choice makes the field fall off
more quickly than either a unipolar magnetic field configuration
or that of the solar active region magnetic field (Gary & Linsky
1981). Figure 7 shows the dependence of magnetic field strength
on height of the radio emission region above the photosphere.
Combining Eqs. 9 and 10, the radio emission region and
therefore the magnetosphere can be constrained in the range of
0.56 R∗ to 6.75 R∗ . A possible position for the radio-emitting region in the radio active UCDs at ∼4.9 GHz is 1.5 R∗ from their
center, 1.06×105 km in the case of TVLM 513. This implies that
the magnetic field strength at the photosphere or chromosphere
would be as large as 7,000 G. More observations in the optical
(Zeeman Doppler effect) and infrared bands are needed to constrain the magnetic topology and the behavior of the plasma in
the magnetic flux tube and to determine whether such large field
strengths exist on these objects.
In addition, we expect the existence of an enhanced ambient wave energy background by gyroresonance turbulence or gyrosynchoroton radiation and some intense events at other wavelengths, e.g. optical or X-ray emission, which would occur from
the process where hot plasma starting from collision-less region
collides with the collisional chromosphere or photosphere. In the
case of X-ray emission, thermal bremsstrahlung emission and
inverse Compton scattering could be the responsible mechanism
since there are hot plasmas with T h up to 108 K and possible low
energy photons. However, since the Thomson scattering optical
depth of the corona of the brown dwarf is 1, the contribution
of inverse Compton scattering is unlikely to be important.
6. Conclusions
An active region model is applied to the radio emission from a
cool dwarf, in which the ECMI mechanism is responsible for
the radio bursts from the magnetic tubes, while the rotation of
the dwarf can modulate the total observed flux with respect to
time. The time profile of the radio light curve is in the form of
power law in our model. Using this model, we can determine
the nature (e.g. size, temperature, density) of the radio-emitting
region plus the magnetic topology can be constrained as well.
In the case of TVLM 513, our model shows the loss-cone
electrons have a density in the range of 1.25 × 105 − 5 × 105
cm−3 and temperature between 107 and 5×107 K. The brightness
temperature is typically ∼ 1015 K for pulses, ∼ 5 × 1010 K for the
background emission, implying the ECMI mechanism operates
in compact region of ∼0.007 RJup if the active region is at 30◦ .
For an active region closer to the pole, e.g. 70◦ , the size is ∼60%
smaller, implying a higher brightness temperature.
The model predicts an enhanced ambient wave energy background and a ≈7000 G surface magnetic field strength. The theoretical X-ray flux density in our model is much smaller than
a marginal X-ray observation of TVLM 513, which implies a
more complicated plasma behavior or magnetic structure on the
dwarf. Additional multi-wavelength observations are needed to
constrain the tentative conclusions and help us to improve the
understanding of the magnetic field on ultracool dwarfs and to
test the viability of this model in comparison with others, such
as the auroral model.
Acknowledgements. The Armagh Observatory is supported by a grant from
the Northern Ireland Dept. of Culture Arts and Leisure. GH and AG gratefully acknowledge the support of Science Foundation Ireland (grant No.
07/RFP/PHYF553). AK, SYU & JGD thank the Leverhulme Trust for support. AA gratefully acknowledges the support of the Scientific Research Fund
of ”St. Kl. Ohridski” University of Sofia (grant No. 80/2009 and 138/2010).
ALM & SYU thank M. Aschwanden for providing the quasi-linear diffusion
code of ECMI. SYU thanks Gavin Ramsay for his comments and also thanks
Yu et al.: Modelling the radio pulses of an ultracool dwarf
Eamon Scullion for discussions. We also thank the UK Science and Technology
Facilities Council for support via a Visitor grant. We gratefully thank the referee
for his/her suggestions and comments.
References
Afram, N., Reiners, A., & Berdyugina, S. V. 2009, in Astronomical Society of
the Pacific Conference Series, Vol. 405, Astronomical Society of the Pacific
Conference Series, ed. S. V. Berdyugina, K. N. Nagendra, & R. Ramelli, 527–
+
Antonova, A., Doyle, J. G., Hallinan, G., Bourke, S., & Golden, A. 2008, A&A,
487, 317
Antonova, A., Doyle, J. G., Hallinan, G., Golden, A., & Koen, C. 2007, A&A,
472, 257
Aschwanden, M. J. 1990a, A&AS, 85, 1141
Aschwanden, M. J. 1990b, A&A, 237, 512
Audard, M., Osten, R. A., Brown, A., et al. 2007, A&A, 471, L63
Berger, E. 2002, ApJ, 572, 503
Berger, E. 2006, ApJ, 648, 629
Berger, E., Ball, S., Becker, K. M., et al. 2001, Nature, 410, 338
Berger, E., Gizis, J. E., Giampapa, M. S., et al. 2008, ApJ, 673, 1080
Berger, E., Rutledge, R. E., Phan-Bao, N., et al. 2009, ApJ, 695, 310
Burgasser, A. J., Kirkpatrick, J. D., Reid, I. N., et al. 2000, AJ, 120, 473
Burgasser, A. J. & Putman, M. E. 2005, ApJ, 626, 486
Burrows, A., Hubbard, W. B., & Lunine, J. I. 1989, ApJ, 345, 939
Cattell, C., Bergmann, R., Sigsbee, K., et al. 1998, Geophys. Res. Lett., 25, 2053
Chabrier, G. & Baraffe, I. 2000, ARA&A, 38, 337
Dahn, C. C., Harris, H. C., Vrba, F. J., et al. 2002, AJ, 124, 1170
Donati, J., Forveille, T., Cameron, A. C., et al. 2006, Science, 311, 633
Dorman, B., Nelson, L. A., & Chau, W. Y. 1989, ApJ, 342, 1003
Doyle, J. G., Antonova, A., Marsh, M. S., Hallinan, G. Yu, S., & Golden, A.
2010, A&A, submitted
Doyle, J. G., Butler, C. J., Bryne, P. B., & van den Oord, G. H. J. 1988, A&A,
193, 229
Dulk, G. A. 1985, ARA&A, 23, 169
Dulk, G. A. & McLean, D. J. 1978, Sol. Phys., 57, 279
Ergun, R. E., Carlson, C. W., McFadden, J. P., et al. 2000, ApJ, 538, 456
Ezoe, Y., Ishikawa, K., Ohashi, T., et al. 2010, ApJ, 709, L178
Gary, D. E. & Linsky, J. L. 1981, ApJ, 250, 284
Gizis, J. E., Monet, D. G., Reid, I. N., et al. 2000, AJ, 120, 1085
Grießmeier, J., Zarka, P., & Spreeuw, H. 2007, A&A, 475, 359
Guedel, M. & Benz, A. O. 1993, ApJ, 405, L63
Hallinan, G., Antonova, A., Doyle, J. G., et al. 2006, ApJ, 653, 690
Hallinan, G., Antonova, A., Doyle, J. G., et al. 2008, ApJ, 684, 644
Hallinan, G., Bourke, S., Lane, C., et al. 2007, ApJ, 663, L25
Jardine, M. & Cameron, A. C. 2008, A&A, 490, 843
Lau, Y. Y. & Chu, K. R. 1983, Physical Review Letters, 50, 243
Leggett, S. K., Allard, F., Geballe, T. R., Hauschildt, P. H., & Schweitzer, A.
2001, ApJ, 548, 908
Leto, G., Pagano, I., Linsky, J. L., Rodonò, M., & Umana, G. 2000, A&A, 359,
1035
Liebert, J., Kirkpatrick, J. D., Cruz, K. L., et al. 2003, AJ, 125, 343
Linsky, J. L. & Gary, D. E. 1983, ApJ, 274, 776
Littlefair, S. P., Dhillon, V. S., Marsh, T. R., et al. 2008, MNRAS, 391, L88
Melrose, D. B. & Dulk, G. A. 1982, ApJ, 259, 844
Melrose, D. B., Dulk, G. A., & Hewitt, R. G. 1984, J. Geophys. Res., 89, 897
Mohanty, S., Basri, G., Shu, F., Allard, F., & Chabrier, G. 2002, ApJ, 571, 469
Morin, J., Donati, J., Forveille, T., et al. 2008, MNRAS, 384, 77
Neuhäuser, R., Briceño, C., Comerón, F., et al. 1999, A&A, 343, 883
Omura, Y., Heikkila, W. J., Umeda, T., Ninomiya, K., & Matsumoto, H. 2003,
Journal of Geophysical Research (Space Physics), 108, 1197
Osten, R. A., Hawley, S. L., Bastian, T. S., & Reid, I. N. 2006, ApJ, 637, 518
Parker, E. N. 1955, ApJ, 122, 293
Pritchett, P. L. 1984, J. Geophys. Res., 89, 8957
Queinnec, J. & Zarka, P. 1998, J. Geophys. Res., 103, 26649
Reid, I. N., Kirkpatrick, J. D., Gizis, J. E., & Liebert, J. 1999, ApJ, 527, L105
Reid, I. N., Kirkpatrick, J. D., Liebert, J., et al. 2002, AJ, 124, 519
Reiners, A. & Basri, G. 2007, ApJ, 656, 1121
Reiners, A. & Basri, G. 2010, ApJ, 710, 924
Rockenfeller, B., Bailer-Jones, C. A. L., Mundt, R., & Ibrahimov, M. A. 2006,
MNRAS, 367, 407
Rutledge, R. E., Basri, G., Martı́n, E. L., & Bildsten, L. 2000, ApJ, 538, L141
Rybicki, G. B. & Lightman, A. P. 1979, Radiative processes in astrophysics, ed.
New York, Wiley-Interscience, 1979. 393 p.
Saur, J., Neubauer, F. M., Connerney, J. E. P., Zarka, P., & Kivelson, M. G. 2004,
Plasma interaction of Io with its plasma torus, ed. Bagenal, F., Dowling, T. E.,
& McKinnon, W. B., 537–560
11
Schmidt, S. J., Cruz, K. L., Bongiorno, B. J., Liebert, J., & Reid, I. N. 2007, AJ,
133, 2258
Stelzer, B., Micela, G., Flaccomio, E., Neuhäuser, R., & Jayawardhana, R.
2006a, A&A, 448, 293
Stelzer, B., Schmitt, J. H. M. M., Micela, G., & Liefke, C. 2006b, A&A, 460,
L35
Tinney, C. G., Mould, J. R., & Reid, I. N. 1993, AJ, 105, 1045
Tinney, C. G., Reid, I. N., Gizis, J., & Mould, J. R. 1995, AJ, 110, 3014
West, A. A., Hawley, S. L., Walkowicz, L. M., et al. 2004, AJ, 128, 426
Willes, A. J. & Wu, K. 2004, MNRAS, 348, 285
Willes, A. J. & Wu, K. 2005, A&A, 432, 1091
Wolszczan, A. & Frail, D. A. 1992, Nature, 355, 145
Wu, C. S. & Lee, L. C. 1979, ApJ, 230, 621
Zarka, P. 1998, J. Geophys. Res., 103, 20159
Zarka, P. 2004, Advances in Space Research, 33, 2045
Zarka, P. 2007, Planet. Space Sci., 55, 598
Zarka, P., Hess, S., & Mottez, F. 2005, AGU Fall Meeting Abstracts, A1275+